Harmonic Addition Theorem

Author

Eric W. Weisstein
January 28, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/HarmonicAdditionTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HarmonicAdditionTheorem.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Identities

a cos(q)+b sin(q)==(Sqrt[(a^2)+(b^2)]) sgn(a) cos((tan^(-1))(-(b/a))+q)

FullSimplify[Subtract@@{a Cos[q]+b Sin[q],Sign[a](Sqrt[(a^2)+(b^2)])Cos[q+ArcTan[-b/a]]},(a|b)∈Reals]